Welcome to another Mathologer video.

Today’s video is about the absolutely

wonderful wobbly table theorem. Some of

you may already be familiar with a very

pretty special case of this theorem

which became quite well known a few

years ago when Numberphile dedicated a

nice video to it. This special case of

the theorem runs as follows: take a

square table like the one over there. At

the moment this table is hovering above the

floor – a bit unusual behavior for a table

but perhaps it’s due to the ghostly

presence at our seance. Anyway the feet of

our spooky table are blue and the points

directly below the four feet are marked

in green. The ghost then disappears and the

table plunks onto the surface. Chances

are that it will wobble, right? Not

because the ghost is still playing games

but just because the floor is uneven. How

do we stabilize the table? Well, usually

people will wedge a napkin or whatever

under one of the wobbling feet, like this.

However, the wobbly table theorem says

there’s another method. The theorem says

that if the floor is not too crazily

uneven, then you can stabilize the square

table simply by rotating the table on

the spot, like in this example. So just

rotate it and eventually all four feet

will be touching the ground. Super neat

isn’t it? And this actually works very

well in practice. Try it on your next

cafe trip for example. And this is only a

small piece of some nice and mostly

unknown table turning maths which also

applies to non square tables. Well let’s

see. Now before we get going I have to

make two very important disclaimers. First when I’m talking about a square

table I always mean that the feet of the

table form a square. Of course, that’s not

always true, even if the table surface is

square. Often enough real-life wobbling

is not caused by the uneven floor but by

the table having feet of unequal lengths.

For example, over there we’ve got a table

on a nice flat surface but one of the

legs has been gnawed at by

a beaver or a ghost or whatever. Anyway

one leg is shorter so even though the

table is standing on a flat ground

obviously there’s no way of shifting the

table around to make it stable. Well we

could turn it over with its legs in the

air but that would be cheating. So, again,

by a square table we mean the feet of

the table form a square. Second

disclaimer. By having stabilized a table

we do not mean that the tabletop is

necessarily horizontal. Usually the

stabilized table will still slope in

some manner consistent with the uneven

floor. Of course such a slope can also be

annoying but definitely not as annoying

as a wobble and typically in real life a

slight unevenness in the ground will

translate into a slight and not too

annoying slope of the tabletop. With

those two disclaimer stored away it’s

time to have some serious fun turning

the tables. Now to do that

we’ll start with Homer Simpson. Of course

it’s never the wrong time to introduce

some Simpsons into the discussion. Recall

that my last video was all about Fourier

series and how to draw complicated

pictures like this Homer drawing with

epicycles. I’ll announce the winner of

the epicycle competition at the end of

this video. For now let me reCYCLE 🙂

Homer as a warm-up exercise for our

table turning. Let’s begin by framing

Homer in a rectangle like this. I call

this sort of rectangle a hugging

rectangle because Homer touches all four

sides of the rectangle. Of course there are

infinitely many hugging rectangles, one

for each possible orientation. Here are a few

more. There’s one and there’s another one

and one more. Now the aspect ratios of

Homer’s hugging rectangles will vary but

it turns out that one of them is special.

One of the hugging rectangles is a

perfect square. And that was not luck. It

turns out that any picture will have a

hugging square. Not obvious at all but

it’s true. We’ll give the idea of a proof in

a minute but first here a couple more

examples to illustrate. Apple, Apple, Apple,

and here is its square. Linux penguin, and

here’s its square. Fish … square. Just a few

random points … square. Now it can happen

that the shape has more than one hugging

square. For example, all hugging

rectangles of circles are squares.

Question for you: Are there any other

shapes all of whose hugging rectangles

are squares?

Leave your answers in the comments.

Here’s a hint. Take a close look at the

Mathologer homepage. Okay here’s the gist

of a proof that every picture has a

hugging square. We start with any hugging

rectangle as in the example over there

and color pairs of opposite sides red

and blue. Now let’s rotate this rectangle

through an angle of 90 degrees so that

at every stage we have a hugging

rectangle. Alright the key observation

is that the dimensions of the hugging

rectangle will change slowly as we

rotate with no sudden jumps. Using maths

jargon we would say that the dimensions

change continuously as we rotate. Now

having rotated through 90 degrees we’re

back where we started, with the same

hugging rectangle. Let’s double check. Hmmm, yep exactly the same right? Well not

quite. Have a closer look. What has

happened is that the red and blue sides

have swapped places. That may not seem

important but it’s the key to capturing

our hugging square. Okay back to the

beginning zero degrees. Let’s record the

difference between the red and blue

lengths as we rotate. To begin red is

longer than blue, so the starting

difference will be what? Positive

negative or zero? Well positive, of course!

However, because of the swapping, at the

end of the 90 degree turn red will be

shorter than blue and so the difference

will be negative. Now since the hugging

rectangle changes continuously this implies that

the difference red minus blue will also

change continuously, … from positive to

negative. And this means that at some

point the difference has to be zero. But

the difference being zero means that red

is equal to blue, that the red sides are

just as long as the blue sides. And, of

course, that means that at this instant

we have a hugging square. And that works

for any picture whatsoever, not just

Homer. Super neat, isn’t it? Just start

with any hugging rectangle, start

rotating and you’re guaranteed to come

across the hugging square. What does all this have

to do with table turning? Well a very

similar above-then-below argument shows

why, as you turn your wobbling square

table through 90 degrees you can expect

to come across a stable position with

all four feet touching the ground. Okay

mathematical seatbelts on? Then here we

go.

Say that square table is on the ground

and it’s wobbling with the two blue feet

touching the ground and the red feet

moving up and down as the table is

rocking back and forth. Now wobble the

table such that the two red feet are

exactly the same vertical distance from

the ground. We’ll call that an equal

hovering position. Now let’s rotate the

table 90 degrees,

always with the two feet in the air in

an equal hovering position. Here we go.

Okay, how have we ended up? Well the table

is exactly in its starting position,

except red and blue have swapped. Now

it’s the red feet on the ground with the

blue feet hovering. But obviously the

swapping is only possible at an instant

where both the blue feet and a red feet

are on the ground. We can also graph this

as we did for the hugging rectangles.

Let’s do it. Here red and blue stand for

the vertical distances of the red and

blue feet from the ground. Then just as

before this difference function starts

are positive and ends up negative

and so should be exactly zero at some

instant along the way. As earlier we

concluded both red and blue are the same

at this point. However, since at least one

of red and blue is zero at all times, at

this special pink zeroing time both the

red and blue distances must be zero. This

means that all four feet are touching

the ground and our table has been

stabilized. Fantastic! Okay, so a square

table can always be balanced by rotating

it. Always? Really? Well, let’s put it

like this: I’ve been using this trick for

decades to stabilize real-life tables

and it’s never failed me. On the other

hand, it’s quite easy to conjure up

scenarios in which different parts of

our argument break down and/or

stabilizing by rotating won’t work.

Second challenge for you today: Try to

come up with some setups that foil our

argument. Just to get things going here’s

something very simple. When we lower the

table over there its top will hit the

top of the mountain and the feet will

never reach the ground. n=Now that we have

the square table sorted give or take

some finicky details let me tell you a

little bit about the history of

mathematical table balancing, the broader

scope of the nifty unwobbling-by-turning argument and the part my friends

and I played in turning this intuitive

argument into a nailed down mathematical

theorem. Okay

hands up who knows this guy. Marty’s

hand is up. Well out there in YouTube

land not many people are nodding their

heads. Marty this is terrible, our hero

has been forgotten. This is Martin

Gardner, by far the greatest popularizer

of mathematics of all time. Starting in

the 1950s and for a quarter of a century

Gardner wrote the hugely influential

Mathematical Games column in Scientific

American. He’s also the author of more

than 100 books on popular mathematics,

magic tricks, and so on. Gardner inspired

more people of my generation to become

mathematicians

than anybody else. For example you can

only watch this video because I really

got into maths as a result of reading

Gardner’s books. And if you enjoy

Mathologer videos it’s mostly because of

the lessons I’ve learned from Martin

Gardner. He was a master at explaining

real mathematics as simply as possible,

and no simpler. In fact much of popular

mathematics in books or on YouTube or

wherever have their origin in a Martin

Gardner column. Hexaflexagons, Conway’s

Game of Life, Dragon fractals, public key

cryptography, wobbly tables. Yes table

turning. In the May 1973 issue of his

Mathematical Games column Gardner

challenged his readers to discover a

version of our heuristic argument for

why stabilizing a square table by

turning should work. He then supplied an

answer to his puzzle in his next column.

Gardner credits Miodrag Novakovic

and Ken Austin as his source for the

wobbly table trick. I actually tracked

down Ken Austin about 15 years ago and

he told me that in fact it was his

friend Miodrag who discovered the trick

and how to justify it around 1950. Ken

only communicated the trick to Martin

Gardner. Miodrag’s version of the table

turning is actually a bit different from

the one I showed you and is also well

worth checking out. Yet another version

of the argument is presented by the

prominent German mathematician Matthias

Kreck who is the star of Numberphile’s

2014 table turning video, also definitely

check out that one. Now for a surprising

twist have a look at the last sentence

of Martin Gardner’s write-up: “A similar

argument can be applied to wobbly

rectangular tables by giving them 180

degree rotations, Now of course that’s

very welcome news since out in the wild

most tables are rectangular but not

square in shape. And there are lots of

other objects whose feet form rectangles

to which all this applies. For example I

personally use stabilizing by rotating

quite often with my trusty stepladder.

Anyway why does everybody only go on

about square tables if this all works

much more generally for rectangular

tables. Well when you try to adapt this

slick argument for square tables to

general rectangular tables you’ll find

that this is not nearly as

straightforward as Gardener’s

off-the-cuff comment suggests. Now before

I show you how to extend the argument to

rectangles

let me tell you about the almost

definitive paper on wobbly tables which

a couple of friends and I published in

2007, in the Mathematical Intelligencer.

There Marty is also taking a bow. This

paper was all about making the wobbly

table argument into a nailed down

mathematical theorem by hammering out on

which surfaces a perfect rectangular

table can be stabilized by turning it on

the spot. One of the main results of our

paper is that stabilizing by turning is

guaranteed to work if the ground and the

table have the following properties:

First the ground. The ground has to be

Lipschitz continuous with associated

angle of at most 35.26 degrees.

Whoa that sounds scary

but it’s not really that bad. All it

means is that the ground is given by a

continuous function and that between any

two points on the ground the ground

slopes at an angle of at most 35.26

degrees. Another way of putting this

is to say that when you slide around the

cony 3d counterpart of the green wedge,

all of the ground will always be below

the wedge. So the scary term Lipschitz

continuous amounts to a maximum slope

requirement for our ground. Okay so

ground continuous and not too steep, check!!!

And what about the table?

Well obviously in line with our

introductory disclaimer the feet should

form a rectangle and we also require a

minimum leg lengths. If the legs are at

least half as long as the diagonals of

the tabletop then nothing can go wrong

in the way of little hills bumping into

the tabletop. So what Bill, Marty, Reiner

and I proved in our paper is that as

long as the surface is continues and not

too sloppy and as long as the table legs

are sufficiently long a perfect

rectangular table can be stabilized by

turning. The heart of this theorem is the

extension of the nice heuristic argument

for square tables but to be able to use

this argument we had to do some pretty

hard work to make sure that our

conditions guarantee that the table

always rotates as nicely as suggested by

the animations that I showed you earlier.

The idea underlying the proof and many

similar proofs is the so-called

Intermediate Value Theorem. This is a

mathematically rigorous version of it

the intuitively obvious fact captured

by our positive-to-negative diagram, the

fact that if a continuous function

changes from positive to negative it

will be 0 somewhere along the way. In

fact, in the proof we use the

intermediate value theorem not just once

but about a zillion times and for those

of you keen on the gory details of the

tricky proof and some of the other nice

results in our paper I’ll provide a link

in the description. Now before we look at

the proof for rectangular tables, here is

an interesting open problem. Is there any

non-rectangular four-legged table which

can always be stabilized by turning. For

example, what about the half-hexagon

table here. So is it possible to

stabilize this or some other mutant

table by rotating? Let me just tell you

about one neat observation. It turns out

that it is important whether or not the

feet of a table lie on a circle. Well

that’s true for our half-hexagon table

and, of course, it’s also true for all

rectangular tables. Why is this circle

business important? Well, it turns out

that if the feet of a table are not on a

circle, then there is definitely a not

too sloppy surface on which this table

will always wobble. Last challenge for

you today, prove this non-concircular-

feet-implies-doomed-to-wobble theorem.

Okay now for the hardcore Mathologer

fans let me sketch

how our simple heuristic rotation

argument for stabilizing square tables

can be extended to rectangular tables.

Okay deep breath. As with square tables

we’ll rotate rectangular tables such that at

all times they are at an equal hovering

position. So at all times throughout the

rotation either the blue pair of feet is

on the ground with the red feet hovering

an equal distance above the ground, as

pictured over there, or vice versa. So if

we can show that at some instant during

our rotation we are guaranteed to have

the blue feet on the ground and there’s

some other instant where we’re sure that the red

feet are on the ground, then we can argue

with continuity as before that somewhere

in between all four feet have to be on

the ground. Now an obvious difference

between the square and rectangular cases

is that a 90 degree turn won’t work for

a general rectangular table. We have to

rotate 180 degrees to bring it back to

the starting position.

But now the problem is that after a 180

degree turn the same pair of feet are on

the ground as at the start. Let’s have a

look. There, turn, same feet on the ground.

So unlike in the square case the end

position after the half turn is not

distinguished in any way from the

starting position.

So our rotation in this case doesn’t

automatically provide us with that

second crucial position where the two

feet that were hovering at the start end

up on the ground. Instead, I’ll show you

that the crucial position always exists

using a proof by contradiction. So let’s

assume that the opposite of what we want

to show is true let’s assume that

somewhere in the universe there’s a

rectangular table and a surface on which

the table cannot be stabilized by

turning. We’re in Fantasyland now and so

to keep things as uncluttered as

possible I’ll only show you the table

and the underlying green points on the

ground. Now

if this table cannot be stabilized by

turning, then the blue feet

must stay on the ground throughout

the rotation and the red feet will be

hovering in the air all the time,

something like that. Alright, blue feet on the

ground and red feed hovering all the

time. Let me now show how this assumption

leads to a contradiction, which then

finishes the proof, modulo all the

finicky details contained in our

Intelligencer paper. Let’s highlight the

rectangle formed by the feet, and let’s

also highlight the z-axis around which

we’re rotating. That brown line is the z-axis. And let’s get rid of the

non-essential bits. Now we make sure that

the center of the rectangle will be on

the z-axis throughout the rotation, like

that. So the center just wanders up and

down the z-axis as we turn. Okay two

important observations. First, since we

are rotating through 180 degrees you’ve

just seen all possible balancing

positions of our table, with the center

on the z-axis. There are no other

balancing positions like this. Second,

let’s call the height of the center of

the table in a certain position the

elevation of the table in this position.

Now, as we rotate, there will be a

position of the table where the

elevation is at a minimum. Let’s see, okay.

so it’s wandering. Now where’s the minimum?

Here, okay, did you spot it? Well, there it

is. We’re almost ready for the punchline.

Notice that if the diagonal connecting

the red points is translated straight

down, the diagonal will connect the green

dots underneath. But this means that we

can rotate the table into a new balancing

position with the blue feet ending up at

those two green points. But this new

balancing position is absolutely

impossible. Why because the elevation of

the new balancing position is lower than

the supposedly minimal elevation that we

started with and that’s the

contradiction. Pretty tricky but also

pretty cool don’t you think?

And that’s it for today as far as table

turning is concerned. I’ve just got one

more thing to do before I sign off,

announce the winner of our epicycle

competition from last time. Lots of nice

ideas and implementations which I’ve

listed in my comment pinned to the top of

the epicycle video comment section. The

one I settled for as the winner is by

Tetrahedri who succeeded in doing

something extra special. Their epicycle

system simultaneously draws out three

iterations of a space-filling curve how

neat is that? Check out the details of

how they did it in the description of

their video. Okay Tetrahedri please send me

your contact details via a comment and

I’ll send you Marty’s and my book. And

thanks again to everybody who took part

in this competition. And that’s it for

today. Happy table-turning.

4:25 intermediate value theorem.

Please say "wobbly" again

19:56 "if this table cannot be stabilised by turning, then the blue feet must stay on the ground throughout the rotation and the red feet will be hovering in the air all the time" it's not obvious why that must be the case. its logical equivalent also says that if you turn the table that way that at at least one position the blue feet don't both touch the ground, it follows that the table can be stabilised which is obviously false

Oh, dear. When I did this, I did it by the ratio of red side to blue side, going from above 1 to below 1. Does that mean I lose a point? (Disclaimer: I did have one professor who did take a point off because 0 was "different" from other numbers so we could use positive and negative but not above and below any other number.)

easier with a round table right? Forget it with a square or rectangular table unless you are lucky, thin and have lots empty space around.

a target, but that would be cheating since the only part the rectangle could hug is essentially a circle.

squares themselves also have this property

Omg this theorem is so useful. Whenever I’m holding a seance where the table has an uneven leg and is on sine waves, this helps so much?

It should be possible to extend this theorem to any 4 legged table, irrespective of the layout of the legs.

Provided that there isn’t an obstacle artificially supporting the table, then 2 opposite legs must be in constant contact with the floor. If rotating the table can cause the other pair of legs to be in constant contact, then the direction of wobble will change.

As long as the floor doesn’t have any sudden level changes (steps), and a rotation can change the direction of wobble, then a rotation can eliminate the wobble irrespective of the individual paths of the legs during rotation.

5:00 Треугольник Рёло? Другие фигуры постоянной ширины?

A square's all hugging rectangle s are squares

Here is a wobbly table cartoon I drew 12 years ago…

http://photos1.blogger.com/blogger/2800/1529/1600/10.jpg

enjoy

Felix Sputnik

I don't feel like this is a full argument, at least for the square table. I think a full argument would be, any 3 points can touch the ground. If you turn it, 3 new points can touch the ground. At some point in that transition, you can have all 4 points touching the ground

Порадовало, что качающийся столик по-английски – "во бля тейбл"

Мне всегда нравилась эта теорема, потому что она имеет важные практические применения в обыденной жизни. Например, я однажды применил её в магазине при покупке стола – он качался, и вращение не помогало его поставить ровно. Аргументируя этим, я заставил продавцов принести другой экземпляр со склада.

a square will have infinite hugging squares

On your t-shirt, the b side should be called the opossum.

Ghost Beavers! RUN AWAY!!!

Where does the 35.26 degree bound come from?

Where does the magical 35.26 angle come from?

Is this related to the Borsuk-Ulam theorem?

The hugging rectangles of all Reuleaux polygons should be square.

tbh any decent waitress knows this…

Intuitively the square case should also have the leg length constraint. Also what exactly is the flaw in your proof that the maximum slope constraint becomes necessary?

hugging rects: equal height shape things like triangle?

If the ground is not continuous or not differentiable, then there's a chance the rotation won't work.

Because a noncontinuous or non differentiable floor means it's possible for the red/blue difference to be non continuous, and thus skip past 0.

An example of a non differentiable floor would be the corner of a sheer cliff on the side of the Grand canyon.

I don‘t understand why we don‘t use 3-legged tables everywhere.

Who else is triggered by the labeling of the triangle's edges on his t-shirt? 😀

Ur logo has only hugging squares

I wonder if it is possible to find out if all legs of the table are the same length?

What if I rotate the table first 180 degrees and if it’s wobbling then in another direction as before, we should have defect legs?

but a picture of a triangle would not have a hugging square, would it?

0:00 Video Starts

1:53 Disclamers

1. Table legs need to bof equald lenghts

2. Table is "stable" in the sense that all parts of the tableis touching the ground

. In real life this is not a big deal as it is hardly noticeable so the application of this

model to real life still holds good

3:48 "Hugging rectangles"/ Intermediate value theorem

6:10 Using calculus to find hugging rectangles of different shapes

7:48 Simulation of the model for squares

8:47 Using "Hugging rectangels" principle to find points of stability

12:10 The different models of table turning , Miordrag's and Martiez Krieg "Bier gartne guy"

14:28 The assumptions (Lipschitz continous and leg lenghts and stuff)

16:15 Intermediate value theorem

17:05 Possibility of other shapes of stablisation

17:30 The reason why it needs to be on a circle

18:00 The full showcase of the model

19:07 Rectangular case

19:34 Proof by contradiction

can you explain the example at 10:00 also the 35.5 degree thing like lipchitz feels alright but 35.5 just seems like some random number

That's all good and nice with the wobbly table, but you might find yourselves being seated a bit uncomfortable after you also level all those pesky wobbly chairs around it :o)

Wouldn't a reuleaux triangle have a square as every hugging rectangle… AND it be the same, stationary square for every orientation of the shape…?

The feet of the table could form a square which is not parallel to the table surface 😉

9:33 but you said yourself that sometimes the issue is that the legs are uneven so surely it has failed sometimes?

I see Tux in that thumbnail

Nie mehr wackelige Tische. Danke für das Video.

At 4:50, aren't those "random dots" the anti-copying constellation used on banknotes?

i make 3 legged tables to avoid this .

A square only has hugging squares.

Where can I get like, your entire closet?

Like srs I get all of your jokes!

Is there a refrigerator door theorem? How do I make my refrigerator slightly un-level in a way that always makes the door try to close by itself due to gravity, no matter what position it is open to? My freezer does this, but my refrigerator is opposite and gravity makes the door try to open more once it's open past a certain point. Instead of randomly adjusting all the feet, it would be more fun to figure out the math on how to make this happen. I also have other doors that I would prefer gravity to tend to open, and yet others I would prefer gravity to tend to close, and even others I would prefer to just stay where I put them no matter what position they are in. I think that this should be possible by adjusting the hinges, perhaps with a shim behind either the upper or lower hinges… but again, just randomly doing things is not as good as figuring out the math behind them.

I figured this out on my own years ago while trying to orient my scale to have all four feet firmly on my non-level floor. I always found that I could eventually find a rotation where all four feet were on the floor.

Hmm… But a triangle cannot have a hugging square, right?

Hippotenuse?

STOP IT

The Long Table Stability Theorem

The Long Term Stable Distrobutoon

Any shape with rotational order of symmetry 2^n, n an integer greater than or equal to 2, will have all its hugging rectangles as hugging squares.

5:00, a square

Came here for Linux thumbnail.

For the rectangle case, you can also do a contraction along one axis of the table and the surface to reduce the rectangular case to the square case.

will all of a squares hugging rectangles be squares?

D: another SQUARE

so maybe an octagon as well? any power of 2 times 4 sided object I guess

I'm pretty sure you can create a disturbance in the continuous surface of the floor that prevents the table from finding a secure position

This morning I was sitting at a wobbly table with my son, and we we're getting increasingly annoyed. I recalled this video, and told my son the wobbles might go away if we rotated the table. After a one eighth counterclockwise turn, the wobble decreased. Another quarter counterclockwise turn and the wobble was gone!

A squares hugging rectangle ate always square

I cant tell if you said "spooky table" or "spoopy table"

I really want it to be the latter

If the floor is perfectly level and its one table leg that is too short, then what?

Nice shirt by the way😀

reminds me of being able to follow any given temperature around the world. i think vsauce did that one

Thank you for the video! All of you friends awesome!

Reuleaux triangle also has all hugging rectangles as squares….

& all other many many types of fixed width shapes as well

Can we please have dark mode? The ultra white background is burning my retina.

Wonderful. Thank you so much.

if all the legs are equal…………. ( edit, I wrote this too fast)

14:32 Ground conditions for this to work. 17:05 Table leg must be on a circle.

I never realised you looked like homer, skinnier though !! Everything makes sens now 😀

Why not make 3 legged stable table able to wobbling not?

9:24 regular polygon shapes

Just get a three-legged table.

A square would have infinite hugging squares that are squares, wouldn't it?

@2:15 curse those Ghost Beavers and their hunger for table legs.

Tux on the thumbnail??

Squares have all their hugging rectangles be squares.

It looks to me as though you're using a vertical projection of the corners of the table to find the contact points. But as soon as you rotate the the surface of the table about some combination of the X and Y axes ("tilting" the table), your corner projection changes from its original aspect ratio to some other aspect ratio. (In the case of the square table, when you tilt the table, the projection goes from square to rectangle.)

I don't know that this causes an insoluble problem for the intermediate value theorem, since all such projections would also be changing continuously, but I'm not sure that the way you've presented the proof here is sufficient to make your case.

1:30 Um, no. IRL the floor is flat and the table legs aren’t in a plane. Never works

The table being able to rotate relies on the assumption that there exists an angle you can turn the table to, at any given position of the table, such that the 2 red legs are equally distant from the ground. Given a sufficiently large dip in the height of the ground, this is no longer true.

All shapes of constant width (ie. Reuleaux triangle, etc…) have hugging rectangles that are all squares.

You simplify math. Amazing. Keep it up

suppose you sit in a beer garden…shapes of constant widths have square hugging-rects?

Where does the 35.26° requirement arise from? Shouldn't it be sufficient for the surface to be continuous for the intermediate value theorem to apply?

What if you have a slightly unravel legged table and an uneaven floor ?

Thanks for reminding me of this! I once had an idea that this could be a way to get the moral equivalent of the Intermediate Value Theorem in an intuitionistic logical framework. I will think about this a bit harder and try to make the vague idea more concrete, ….

For the hugging square puzzle, Rouleaux "polygons" (I don't know how else to call them) are another possibility. Of course, regular squares also have only hugging squares, as opposed to hugging rectangles.

Do you have any idea how can I implement an animation in GeoGebra that rotates a rectangle, always touching the image, like the one you have in the video? (Always beeing a hugging rectangle)

https://youtu.be/aCj3qfQ68m0?t=322

Oh how the tables have turned… *ba dum tsssss

I think your going for the fact that any curve of constant width has a hugging square that you can rotate about it like a circle, which I learned from a different video of yours. Only a year and a bit late to make this comment!

It's tux

Amazing video 😍✌❤️❤️

Do you have a uecix network email address for consultations?

I knew who Martin Gardner was, just never saw his face 🙂

This type of geometry depended on the development of computer graphics allowing continuous changes in diagrams.

The theorem of plan, tells: "The table of three foots, stays more better than with four foots! …

the proof of hugging square is very similar to the fried egg theorem

non circular feet wobbling counter example is trivial, simply have a circular groove on the leg outside the circle at a different height to the other legs, but constant. No matter how you rotate it the hight difference will always cause a wobble.

Previously, on You Know What's BS….

https://www.youtube.com/watch?v=VzdAJJ4JnVM

God forbid you have a wobbly rectangular table up against a wall

All hugging rectangles of circles and squares are square, but also octogons and any regular poligon with 4*n+4 sides. For example a 12-gon.

You want proof? Eat my shorts.

EDIT: actually, any shape with 4-way symmetry.

Next time the desk is wobbly I'll whip out the calculator and show it who's boss